| import OAI.Analysis.FiniteFactor.OrbitSampling.Error |
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| namespace OAI |
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| namespace FiniteFactor |
| open MeasureTheory Filter |
| open scoped Classical Topology |
| noncomputable section |
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| /-- The distinct and mixed error budget is summable along the actual stages. -/ |
| theorem sourceSamplingCorrelationBudget_summable : Summable sourceSamplingCorrelationBudget := by |
| apply (sourceSampling_inverse_square_summable.mul_left (208*Real.exp 135)).of_nonneg_of_le |
| · intro stage; unfold sourceSamplingCorrelationBudget; positivity |
| intro stage |
| let m : ℝ := sourceBlockSize stage |
| have hm : 0 < m := sourceBlockSize_pos stage |
| have hm1 : 1 ≤ m := by |
| have hh : 100 ≤ sourceBlockSize stage := (sourceStage stage).m_ge_100 |
| dsimp [m] |
| exact_mod_cast (show 1 ≤ sourceBlockSize stage by omega) |
| have hd : ((sourceSamplingPrefix stage).card : ℝ) ≤ 2*m := |
| (sourceSamplingPrefix_card_bound stage).trans (by dsimp [m]; nlinarith) |
| have he : m*m^(-74 : ℝ) = m^(-73 : ℝ) := by |
| conv_lhs => lhs; rw [← Real.rpow_one m] |
| rw [← Real.rpow_add hm]; norm_num |
| calc |
| sourceSamplingCorrelationBudget stage ≤ (2*m)*(104*m^(-74 : ℝ))*Real.exp 135 := by |
| unfold sourceSamplingCorrelationBudget |
| exact mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right hd (by positivity)) (by positivity) |
| _ = (208*Real.exp 135)*m^(-73 : ℝ) := by rw [← he]; ring |
| _ ≤ (208*Real.exp 135)*m^(-2 : ℝ) := mul_le_mul_of_nonneg_left |
| (Real.rpow_le_rpow_of_exponent_le hm1 (by norm_num)) (by positivity) |
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| /-- Summable expectations for the full empirical sampling error. -/ |
| theorem sourceSamplingSquaredError_summable_integral (r : Bool) (b : SourceCoordinate → Bool) |
| (a c a' c' : SourceCoordinate → ℝ) : |
| Summable (fun stage => ∫ θ, sourceSamplingSquaredError stage r b |
| (fun i => Set.Icc (a i) (c i)) (fun i => Set.Icc (a' i) (c' i)) θ ∂sourceAngleLaw) := by |
| have hd := sourceSamplingDiagonalContribution_summable_integral r b |
| (fun i => Set.Icc (a i) (c i)) (fun i => Set.Icc (a' i) (c' i)) |
| (fun _ => measurableSet_Icc) (fun _ => measurableSet_Icc) |
| have hb := hd.add (sourceSamplingCorrelationBudget_summable.mul_left 3) |
| apply hb.of_norm_bounded_eventually_nat |
| filter_upwards [sourceBlockSize_tendsto.eventually (eventually_ge_atTop 200000)] with stage hs |
| rw [Real.norm_eq_abs,abs_of_nonneg (integral_nonneg (sourceSamplingSquaredError_nonneg stage r b _ _))] |
| exact sourceSamplingSquaredError_expectation stage hs r b a c a' c' |
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| /-- The full squared sampling error is summable and converges to zero on a |
| conull set of actual source angles. This is not just the self-pair event. -/ |
| theorem sourceSamplingSquaredError_ae (r : Bool) (b : SourceCoordinate → Bool) |
| (a c a' c' : SourceCoordinate → ℝ) : |
| ∀ᵐ θ ∂sourceAngleLaw, |
| Summable (fun stage => sourceSamplingSquaredError stage r b |
| (fun i => Set.Icc (a i) (c i)) (fun i => Set.Icc (a' i) (c' i)) θ) ∧ |
| Tendsto (fun stage => sourceSamplingSquaredError stage r b |
| (fun i => Set.Icc (a i) (c i)) (fun i => Set.Icc (a' i) (c' i)) θ) atTop (𝓝 0) := by |
| have he := sampling_ae_summable_of_integrals sourceAngleLaw |
| (fun stage θ => sourceSamplingSquaredError stage r b (fun i => Set.Icc (a i) (c i)) (fun i => Set.Icc (a' i) (c' i)) θ) |
| (fun stage => sourceSamplingSquaredError_measurable stage r b _ _ (fun _ => measurableSet_Icc) (fun _ => measurableSet_Icc)) |
| (fun stage => sourceSamplingSquaredError_integrable stage r b _ _ (fun _ => measurableSet_Icc) (fun _ => measurableSet_Icc)) |
| (fun stage θ => sourceSamplingSquaredError_nonneg stage r b _ _ θ) |
| (sourceSamplingSquaredError_summable_integral r b a c a' c') |
| exact he.mono (fun θ hθ => ⟨hθ,hθ.tendsto_atTop_zero⟩) |
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| /-- One source-angle event handles any countable family of interval tests, |
| orientations and triangular-coordinate choices for the full error. -/ |
| theorem sourceSamplingSquaredError_countable_event {ι : Type*} [Countable ι] |
| (r : ι → Bool) (b : ι → SourceCoordinate → Bool) |
| (a c a' c' : ι → SourceCoordinate → ℝ) : |
| ∀ᵐ θ ∂sourceAngleLaw, ∀ j, |
| Summable (fun stage => sourceSamplingSquaredError stage (r j) (b j) |
| (fun i => Set.Icc (a j i) (c j i)) (fun i => Set.Icc (a' j i) (c' j i)) θ) ∧ |
| Tendsto (fun stage => sourceSamplingSquaredError stage (r j) (b j) |
| (fun i => Set.Icc (a j i) (c j i)) (fun i => Set.Icc (a' j i) (c' j i)) θ) atTop (𝓝 0) := |
| ae_all_iff.mpr (fun j => sourceSamplingSquaredError_ae (r j) (b j) (a j) (c j) (a' j) (c' j)) |
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| end |
| end FiniteFactor |
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| end OAI |
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